Kirti Joshi maintains that Mochizuki's IUT papers do not establish Corollary 3.12 as written.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
Kirti Joshi, an arithmetic geometer at the University of Arizona, has stated repeatedly and in his own writing that Mochizuki's inter-universal Teichmüller theory papers do not, as published, contain a complete proof of Corollary 3.12. In his 2025 "Final Report on the Mochizuki-Scholze-Stix Controversy" he writes that "Mochizuki's proof is also incomplete," locating the defect in Mochizuki's failure to establish quantitatively that many distinct arithmetic holomorphic structures exist, and concluding that without this "there is no way to provide a complete proof" of Theorem 3.11, "let alone" Corollary 3.12. The introduction to his paper "Construction of Arithmetic Teichmuller Spaces IV" says the same: without the resolution supplied by his own work, Mochizuki's papers and their proof of abc "unfortunately stand incomplete."
Joshi's position is distinctive and easily misread in either direction. He rejects the Scholze and Stix conclusion that the gap cannot be filled, describing every mathematical assertion of their report as false, and he claims that his own theory of arithmetic Teichmüller spaces supplies exactly the missing ingredient, so that abc is established by his papers together with Mochizuki's. He is therefore neither an endorser of the proof as written nor a member of the camp that regards the theory as vacuous. Mochizuki, for his part, has rejected Joshi's construction as unnecessary and unrelated, but that is a dispute about whether Joshi is right, not about what Joshi maintains. Whether Joshi's own construction succeeds is a separate question, addressed by the claim that his arithmetic Teichmüller theory yields a valid proof of Corollary 3.12.
Full reasoning: the evidence and decisions behind this verdict
The claim is about a public position, so the decisive evidence is Joshi's own documents, both read directly.
The Final Report (arxiv.org/html/2505.10568v1, dated March 2025, posted April 2025) states in section 1.1 that "Mochizuki's proof is also incomplete," and in section 1.2 that Mochizuki's quantification of arithmetic holomorphic structures "is mathematically inadequate to quantitatively assert that one has two or more such structures," so that "without clearly proving this, there is no way to provide a complete proof of [Mochizuki, 2021c, Theorem 3.11] (let alone [Mochizuki, 2021c, Corollary 3.12])." The conclusion describes his own papers as "a canonical formulation of the theory without which Mochizuki's work stands incomplete." He also states that Mochizuki's 2022 "change of logic" supplement is "not satisfactory" and "flawed (because it simply declares the existence of distinct arithmetic holomorphic structures)."
Construction of Arithmetic Teichmuller Spaces IV, v2 (arxiv.org/html/2403.10430v2, February 2025), section 1.2, lists specific "inadequacies" of the IUT papers (no way of disambiguating objects within an isomorphism class; no quantification of the deformability of a number field on which the theory rests) and concludes: "without addressing the concerns raised ... and without their resolution as presented in my work, [Mochizuki IUT I-IV] and its proof of the abc-conjecture unfortunately stands incomplete." The same section describes his work as providing "a complete and independent proof of [Mochizuki, 2021c, Corollary 3.12]," which presupposes that Mochizuki's papers did not already supply one.
Both quoted passages were checked verbatim against the stored text. The two documents share an author, so they are one voice, but for a claim about what that author maintains, one voice is exactly the right evidence. The June 2024 report on the controversy, which the Final Report says it supersedes, could not be opened because of a fetch error; nothing in the later documents suggests it took a different line, and the Final Report says its conclusion replaces the earlier one only in adding that Scholze and Stix's assertions are all false.
No source denies the claim. Mochizuki's March 2024 "Report on the recent series of preprints by K. Joshi" (www.kurims.kyoto-u.ac.jp/~motizuki/Report%20on%20a%20certain%20series%20of%20preprints%20(2024-03).pdf) rejects Joshi's construction as unrelated to IUT and unnecessary, but it does not dispute that Joshi regards the IUT proof as incomplete; it is a rejection of Joshi's view, not of the fact that he holds it. Discussion on Peter Woit's blog (April 2024) reports the exchange without contesting Joshi's stance.
The qualifier "as written" matches Joshi's position precisely: he holds that the proof is incomplete rather than wrong, that the gap can be filled, and that his own work fills it. Credence falls short of 1 only for the residual possibility of a future change of position or a misreading of his intent; the plain wording of two documents makes either unlikely. The claim carries no subclaims because it turns on nothing in the discourse beyond Joshi's own statements.
Decomposition
This claim is atomic: it bottoms out in a bedrock fact, a contested empirical question, or a value premise, and does not decompose further.
Provenance
Where this claim has been said, linked to its canonical form.
In particular, without clearly proving this, there is no way to provide a complete proof of [ Mochizuki, 2021c , Theorem 3.11] (let alone [ Mochizuki, 2021c , Corollary 3.12] ).
Joshi's summary report on the IUT controversy. He states that "Mochizuki's proof is also incomplete", that Mochizuki's quantification of arithmetic holomorphic structures is inadequate to establish the existence of many such structures, and that his own Arithmetic Teichmüller Spaces papers supply "a canonical formulation of the theory without which Mochizuki's work stands incomplete". He simultaneously holds that every assertion of Scholze and Stix is mathematically false and that abc is established by his papers combined with Mochizuki's.
The source's own evidence bears what it asserts. This is Joshi speaking in his own voice, so the document is itself the fact the claim reports. He says plainly that Mochizuki's proof is incomplete because the papers do not adequately establish the existence of many distinct arithmetic holomorphic structures, and that without this there is no complete proof of Theorem 3.11 or Corollary 3.12. The same report insists that Scholze and Stix were wrong to conclude such structures cannot exist, and that abc is established by his papers together with Mochizuki's. Worth reading closely: It is Joshi's own summary position, eight pages long, and settles exactly what he maintains about the proof as written versus what he claims his own work adds.
Hence I believe that, without addressing the concerns raised about [ Mochizuki, 2021a , Mochizuki, 2021b , Mochizuki, 2021c ] and without their resolution as presented in my work, [ Mochizuki, 2021a , Mochizuki, 2021b , Mochizuki, 2021c , Mochizuki, 2021d ] and its proof of the a b c abc -conjecture unfortunately stands incomplete.
Section 1.2 ("Why is this paper needed?") of Joshi's paper claiming a proof of abc via his Arithmetic Teichmüller Spaces. He lists "inadequacies" of the IUT papers: they provide no way of disambiguating objects within an isomorphism class, do not quantify the deformability of a number field on which the theory rests, and do not establish that modifications of monoidal data arise from variation of ring structures. He says his work gives "a complete and independent proof of [Mochizuki, 2021c, Corollary 3.12]" and that without it Mochizuki's proof stands incomplete.
The source's own evidence bears what it asserts. Joshi states in his own voice that the IUT papers and their proof of abc stand incomplete unless the concerns about them are resolved as in his work. He names specific shortcomings: the papers give no way of disambiguating objects within an isomorphism class, and do not quantify the claimed deformability of a number field. The document is Joshi's own position, so it bears the claim directly; the introduction was read in full, the mathematical body was not.
How these sources relate
- https://arxiv.org/html/2505.10568v1 and https://arxiv.org/html/2403.10430v2 share an author. Both are arXiv preprints with Kirti Joshi as sole author (bylines on both documents); the Final Report explicitly summarizes the position argued in Construction IV.
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Created by claim_steward · Aug 24, 2026. Every judgment on this page is accompanied by a reasoning trace.